Apparatus and method for modeling all matter by modeling, truncating, and creating N-dimensional polyhedra including those having holes, concave attributes, and no symmetry

ABSTRACT

An apparatus and method for modeling all matter by modeling, truncating, and creating general polyhedra without requiring advanced computer processors and computer memory according to one embodiment includes at least one processor, means for inputting vertex data to the processor, a display in data communication with the processor, and computer memory coupled to the processor. The computer memory has recorded within it machine readable instructions for storing vertex data previously input to the processor, truncating a polyhedron created using the vertex data, and actuating the display. The instructions for creating and truncating a polyhedron utilize length data and angle data for triangles formed from the vertex data.

RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent applicationSer. No. 11/686,496, filed Mar. 15, 2007 now abandoned the disclosure ofwhich is incorporated herein by references.

BACKGROUND OF THE INVENTION

It has been theorized that a single simple rule generates all knowncomplexity demonstrated by matter. Nevertheless, matter is currentlydescribed and depicted using many complex formulas. While drafting andgeometrical computer programs have become commonplace, they rely onnumerous complex formulas and require advanced computer processors andcomputer memory, and they cannot be used to model, truncate and createall “General Polyhedra”.

A simple computational method for modeling, truncating and creating“General Polyhedra” that allows one to model and study any form in ndimensions has been needed. N-dimensional forms are created from lines,polygons, and polyhedra. For example, a 2-dimensional polygon has1-dimensional sides, a 3-dimensional polyhedron has sides or faces whichare 2-dimensional polygons, a 4-dimensional polyhedron has sides thatare 3-dimensional polyhedra, etc. A 4-dimensional polyhedron that hasfour sides, for example, could be constructed by building a3-dimensional “core” polyhedron, and building four “side” 3-dimensionalpolyhedra that represent the sides of the 4-dimensional polyhedron. Each“side” 3-dimensional polyhedron has a face with dimensions that aremirror image with one face of the “core” 3-dimensional polyhedron, etc.

Old methods have named polyhedra and grouped them into specialcategories depending on their characteristics. A “General Polyhedron”may be referred to in this new method simply as “Polyhedron” with“Polyhedra” for plural, has n dimensions where n faces can have a commonedge, can be concave or convex and can have holes or no holes, and canhave symmetry or no symmetry. That represents all matter.

Old Art to model, truncate, and manipulate polyhedra has had limitationsthat do not generalize to all general polyhedra. First, the definitionof polyhedra has been unsatisfactory. Polyhedra have been named andgrouped into categories based on some common characteristics that theyshare. A defining characteristic of almost all kinds of polyhedra hasbeen that just two faces join along any common edge. A polyhedron hasbeen a 3-dimensional example of the more general polytope in any numberof dimensions. Then, truncation of the polyhedron as defined in MathWorld has been the removal of portions of solids falling outside a setof symmetrically placed planes. Truncation displaces points along theedges of a polyhedron by a ratio r less or equal to ½, where r is thefraction of the edge length at which to truncate, and then constructingthe new polygons. And rectification is the process of truncating apolytope by marking the midpoints of all its edges, and cutting off itsvertices at those points.

Another prior art method, described in U.S. Pat. No. 5,428,717 toGlassner, can truncate only a simply connected concave object with noholes, by transforming it first to its convex hull by popping andsliding the concave edges, identifying midpoints of edges to be used forcomputations and then truncating the convex hull using computations ofintersecting moving planes and half spaces inspired by the classicalconstruction of dual polyhedra. The dual of a Platonic solid orArchamedean solid, for example, can be computed by connecting themidpoints of the sides surrounding each polyhedron vertex, andconstructing the corresponding tangential polygon. Then the truncatedconvex hull will have to be converted again to the concave goal shape.The Glassner method requires extensive computation and has its obviouslimitations.

Another prior art method, described in U.S. Pat. No. 6,396,494 toPittet, frames the object to be studied with known geometric shapes oftheir choice like the tetrahedra that form a cube, then slices saidgeometric shapes using computations of intersecting planes. To speed upsuch very complex computations that require special quick hardware suchas processors and computer systems, labor intensive information for alimited number of slices, 16 predetermined slices in this instance only,are computed by the operator first, then pre-entered into the system tobe used, which leads to limited possibilities and missed information.

The invention disclosed herein uses simple mathematical computations ofangles and side lengths of triangles to model and truncate anyn-dimensional General Polyhedron, not limiting truncation up to themidpoint of edges of the polyhedron, and not using complex calculationsof intersecting planes or half planes, or intersecting moving planes andhalf spaces. Using this new method, and changing the value of a fewvariables, one can quickly truncate any polyhedron along any edge by anyproportion possible of that edge, and at any angle possible adinfinitum, without disregarding any information.

More particularly, using this novel simple computational method, one canmodel a Polyhedron. Then one can truncate or slice or clip any vertex ofsaid polyhedron at any angle or in parallel, and along any proportion ofeach and every edge of said polyhedron that meet at said vertex from 0%and up to the maximum amount of truncation possible. Truncating saidpolyhedron would create new faces of the polyhedron that can be measuredand built thus creating an infinite amount of new polyhedra. Then onecan model another one or more new polyhedron with one face that hasdimensions that are mirror image with either an old face of thepolyhedron, or with a new face created by the truncation. Having the oldand new polyhedron share a face will create one more new polyhedron. Andso on and so forth, one can imagine the infinite complexity that can begenerated and studied. For example, one can tessellate n-dimensionalspaces by filling them with n-dimensional polyhedra.

One can also model a new polyhedron with all its faces having mirrorimage dimensions of the faces of the original polyhedron, thus beingable to depict a R handed polyhedron and a L handed polyhedron forexample, which will facilitate study.

Using this novel method, the polyhedron built will help us be able tomodel, understand and manipulate structures at the nano scale for thestudy of molecular biology for example, or bioinformatics, organicchemistry, crystallography etc; and structures at the macro scale tobuild objects that can be perceived by our senses; and structures atsuch large scale that our senses cannot perceive. And since physicsdescribes matter as the geometric distribution of forces in space, thenthe study and use of this method that uses only angles and side lengthsof triangles may help us understand these forces. The geometricdistribution of forces also helps us understand the other structures inlife, like the dynamics in a family or corporation. In a family, forexample, the special bond between the father and daddy's little girl,the special bond between the mother and the son, and the special bondbetween the father and the mother, define a triangle of their own. Thusthe triangle is the trinity that is the one that is the everything.

The present invention will become obvious after the followingdescription.

SUMMARY OF THE INVENTION

One method disclosed herein for modeling and truncating polyhedraincludes the steps:

-   -   Providing vertex data.    -   Establishing lines with known lengths between the vertices.    -   Establishing triangles with known angles from the established        lines. Some of these triangles form triangular faces of the        vertices of the polyhedron, and the other triangles are        polyhedron virtual triangles.    -   Identifying which triangles belong to the same polygon.    -   Forming polygons that define the faces of the polyhedron using        the triangular faces that may overlap.    -   Identifying the triangular faces of each vertex.    -   Identifying the polyhedron virtual triangles of each vertex.    -   Identifying the triangular faces of the polyhedron as forming        the boundary of the polyhedron.    -   Identifying the triangular faces and the polyhedron virtual        triangles that define the boundary shape of the polyhedron.    -   Identifying the triangular faces and the polyhedron virtual        triangles that extend inside the boundary shape of the        polyhedron.    -   Identifying the adjacent vertices of each vertex to be        truncated,    -   Establishing truncating bases for each vertex to be truncated.    -   Forming truncating base triangles that are used to compute the        truncating bases using each three vertices adjacent to the        vertex to be truncated that are adjacent to each other and that        belong to the boundary shape of the polyhedron.    -   Referring to these vertices as adjacent vertices.    -   Forming two of the truncating base lines of the vertex to be        truncated using two sides of each of the truncating base        triangles.    -   Noting that triangular faces and polyhedron virtual triangles of        the vertex to be truncated that extend inside the boundary shape        of the polyhedron intersect at determined intersecting angles        and intersecting lines with the established truncating base        triangles. Determine which triangular faces and which polyhedron        virtual triangles intersect with which truncating base triangle.    -   Calculating all the intersecting angles and intersecting lines        for all truncating base triangles and their intersecting        triangular faces and polyhedron virtual triangles.    -   Calculating the points where the intersecting lines intersect        with the rest of the edges of the Vertex to be truncated.    -   Calculating the lengths of the rest of the truncating base lines        that complete the truncating bases for each vertex to be        truncated using the intersecting angles and distances generated        by the intersecting lines.    -   Forming truncating virtual triangles that have sides with known        lengths and angles from the truncating base lines.    -   Forming truncating polygons with the truncating base lines as        its sides for the truncating bases.    -   Forming new truncating lines to truncate parallel to the        truncating bases; forming new truncating lines to truncate at an        angle to the truncating bases; or forming new truncating lines        to truncate parallel to truncation at an angle to the truncating        bases.    -   Establishing the dimensions of the new truncating virtual        triangles with known side lengths from the new truncating lines.    -   Forming new truncating polygons from the new truncating        triangles.    -   Establishing the real truncating base that intersects with all        the edges of the vertex to be truncated at either one of its        truncating bases if said truncating base does not intersect with        all the edges to be truncated when it intersects with the line        extension of some of those edges. Establishing said real        truncating base at an angle to said truncating base.    -   Optionally truncating the vertices of the polyhedron at either        one of their truncating bases when it intersects with all the        edges to be truncated.    -   Optionally designating the truncating bases that intersect with        all the edges to be truncated as real truncating bases only for        the purpose of truncating at said truncating bases and for the        purpose of determining limits on truncation.    -   Optionally truncating the Vertex to be truncated at one or more        of its real truncating bases knowing that truncating at one real        truncating base will limit the amount of truncation at the other        real truncating bases.    -   Determining the limits on truncation by lengths of edges to be        truncated and values of the angles that determined the real        truncating bases and limits based on the amount of previous        truncation at said vertex and at adjacent vertices.    -   Optionally displaying the results of the truncation by        displaying the information that belongs to the polygonal faces        of the polyhedron only.    -   Optionally displaying the results of the truncation on a        polyhedral net.    -   Optionally displaying the results of the modeling and truncating        by displaying the faces of the polyhedron separate from each        other.    -   Optionally, one can get vertex and line length data of an object        from real life, then display that information, and then proceed        to truncate this object using the method described above.

An apparatus for creating and truncating general polyhedra without usingcomplex formulas and without requiring advanced computer processors andcomputer memory according to one embodiment includes at least oneprocessor, means for inputting vertex data to the processor, a displayin data communication with the processor, and computer memory coupled tothe processor. The computer memory has recorded within it machinereadable instructions for storing vertex data previously input to theprocessor, truncating a polyhedron created using the vertex data, andactuating the display. The instructions for creating and truncating apolyhedron utilize length data and angle data for triangles formed fromthe vertex data.

An apparatus for creating and truncating general polyhedra according toanother embodiment includes at least one processor, means for inputtingvertex data to the processor, a display in data communication with theprocessor, and computer memory coupled to the processor. The computermemory has recorded within it machine readable instructions for storingvertex data previously input to the processor, truncating a polyhedroncreated using the vertex data and actuating the display. Theinstructions for creating and truncating a polyhedron utilize lengthdata and angle data for triangles formed from the vertex data and do notrequire advanced computer processors and advanced computer memory.

An apparatus for determining properties of general polyhedra withoutusing complex formulas and without requiring advanced computerprocessors and computer memory according to one embodiment includes atleast one processor, means for inputting vertex data to the processor, adisplay in data communication with the processor, and computer memorycoupled to the processor. The computer memory has recorded within itmachine readable instructions for storing vertex data previously inputto the processor, the vertex data being associated with polyhedra,creating triangles using the vertex data, determining at least oneproperty of the triangles, determining at least one property of thepolyhedra using the at least one determined property of the triangles,and actuating the display.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a perspective view of a polyhedron according to an embodiment.

FIG. 2 is a polyhedral net of the polyhedron of FIG. 1.

FIG. 3 is the polyhedral net of FIG. 2 after being modified according toan embodiment.

FIG. 4 is the polyhedral net of FIG. 2 after being modified according toan embodiment.

FIG. 5 is the polyhedron of FIG. 1 after being modified according to anembodiment.

FIG. 6 is another polyhedron after being modified according to anembodiment.

FIG. 7 schematically shows elements of an apparatus for creatingtruncated polyhedra according to an embodiment.

DESCRIPTION OF THE PREFERRED EMBODIMENT

An apparatus and method according to the present invention will now bedescribed in detail with reference to FIGS. 1 through 7 of theaccompanying drawings.

According to embodiments disclosed herein, all truncated polyhedra canbe generated using the simple mathematics of calculating angles and sidelengths of triangles. The infinite amount of truncated polyhedra thusgenerated may form an infinite amount of n dimensional polyhedra,including all of the geometric complexity.

The boundary shape of the polyhedron is defined as being formed by someof the triangular faces of the polyhedron and some of the polyhedronvirtual triangles, while the other polyhedron virtual triangles extendinside the volume of the boundary shape of the polyhedron.

The boundary of the polyhedron is defined as being formed by thetriangular faces of the polyhedron, where some triangular faces of theboundary of the polyhedron can also extend inside the volume of theboundary shape of the polyhedron as in the case of an n dimensionalpolyhedron for example where more than 2 triangular faces can have acommon edge.

FIG. 1 shows a polyhedron 101 constructed using method disclosed herein.At a first step of the method, the three-dimensional coordinates of thevertices (A, B, C, D, V, W) are provided. The method then proceeds to asecond step, where the length of every line segment connecting thevertices (A, B, C, D, V, W) is determined using simple algebra. Thoseline segments form triangles (ABC, ABD, ABV, ABW, ACD, ACV, ACW, ADV,ADW, AVW, BCD, BCW . . . ). In one embodiment, some of these trianglesform triangular faces of the vertices, and the Boundary of thepolyhedron 101 (CWD, AWB, AWD, BWC, AVB, BVC, CVD, DVA) while the restof these triangles form polyhedron virtual triangles. Adjacent verticesmay have overlapping triangles. For example, vertex A has triangularface AVB overlapping the triangular face BVA of vertex B. The rest ofthe triangles may intersect. For example, triangle BVD intersects withtriangle ABC. Putting all of the triangles together provides all thetriangular faces of the polyhedron 101 and all of its vertices and allof the polyhedron virtual triangles. The method then proceeds to a thirdstep, where all of the angles of all of the triangles are calculated.The method then proceeds to a fourth step, where it is determined whichtriangles belong to the same polygon. Thus, for example in the case oftriangle VBW, if angle VBW is equal to angles VBA plus ABW, thentriangle VBW belongs to the same polygon VBWA which will be aquadrilateral. Herein triangle VBW will partially overlap triangles AVBand ABW. In the case of FIG. 1, angle VBW is smaller than the sum ofangles VBA and ABW, which results in VAB, ABW being two triangular facesof the polyhedron, and triangle VBW being a polyhedron virtual triangle.Notably, by determining the properties (e.g., dimensions, etc.) of thetriangles that form the faces of the polyhedron, the dimensions of thefaces of the polyhedron and surface area may be easily determined. Themethod then proceeds to a fifth step, where a vertex of the polyhedron101 may be truncated.

At the fifth step of the method, at least one of the vertices istruncated (e.g., V) by forming all its possible truncating bases. Whilethe truncation of vertex V at one of its truncating bases is describedin great detail here, it should be appreciated that vertex V may betruncated at any of its other truncating bases, and a different oradditional vertex may be truncated following the steps set forth herein.To truncate the vertex, one of the truncating bases is established. Forexample, the truncating base is established at 3 Vertices A, B, C thatare adjacent to vertex V and adjacent to each other to form thetruncating base triangle ABC, which will help calculate the truncatingbase and two truncating base lines AB and BC. That truncating baseintersects edge VD at point X. To establish point X, we calculate theintersecting angle between edge VB and line BL, which is formed wherepolyhedron virtual triangles VBD and ABC intersect. This is angle VBX.

Point L is located where BX intersects AC, and angle VBX is equal toangle VBL. The intersecting angle is then determined using the knownangles ABC, ABD, and DBC in polyhedron virtual triangles ABC, ABD andDBC. The ratio of angle ABD to angle DBC is equal to the ratio of angleABL to angle LBC. Angle LBC is equal to angle ABC minus angle ABL.Therefore, ABD/DBC is equal to ABL/(ABC-ABL), and angle ABL iscalculated. Knowing angles ABL and BAC and length AB in virtual triangleABL, simple algebra and/or trigonometry is used to calculate the lengthsof BL and AL. Because the lengths of AL and AV are known and angle VALis equal to angle VAC in virtual triangle VAL, simple algebra and/ortrigonometry is used to calculate angles ALV and AVL and length VL.Knowing lengths VL and VB and BL in virtual triangle VBL, we calculateangle BLV and the intersecting angle VBL which is equal to VBX. Becauseangle VBL is equal to angle VBX, angle BVX is equal to angle BVD, andlength BV in virtual triangle VBX is known, we calculate length BX andlength VX and thus establish X.

Because lengths VX and VC are known and angle CVX is equal to angle CVDin triangle CVX, we calculate the length of truncating base line CX.Because lengths VX and VA are known and angle AVX is equal to angle AVDin triangle AVX, we calculate the length of truncating base line XA.Thus, together with truncating base lines AB and BC, we establish thetruncating base ABCX. If length VX is greater than length VD, the realtruncating base A′B′C′D that is at an angle to ABCX and intersects VD atpoint D must be established. Truncation could then continue at parallelor at an angle to ABCD taking in account the restrictions imposed by thelength of VD.

Vertex V is then truncated in one of the following ways. It can betruncated at the truncating base by determining the length of BX, whichis the third edge of the virtual triangle BVX and also the third edge ofthe virtual triangles BCX and BAX. Or, it can be truncated at a planeparallel to the truncating base by determining new truncating lines(A1)(B1), (B1)(C1), (C1)(X1), and (X1)(A1). This provides new truncatingpolygon (A1)(B1)(C1)(X1). Each two of these new truncating lines(A1)(B1), (B1)(C1), (C1)(X1), and (X1)(A1), belong to a new truncatingvirtual triangle. New truncating virtual triangles (A1)(B1)(X1), and(B1)(X1)(C1) have a 3rd side (B1)(X1) that is parallel to BX and is alsothe 3rd side of another virtual triangle (B1)V(X1) formed by twonon-adjacent edges of the vertex to be truncated. New truncating virtualtriangles (A1)(B1)(C1), and (A1)(X1)(C1) have a 3rd side (A1)(C1) thatis parallel to AC and is also the 3rd side of another virtual triangle(A1)V(C1). Suppose we start truncating at (A1) with known length V(A1)at a plane parallel to ABCX. (A1)(C1) and (B1)(X1) are lines parallel toAC and BX. They intersect with line VL at (L1). Since angles AVL and BLVare known quantities as calculated above, we can easily calculate anglesVLC as 180-ALV and VLX as 180-BLV. These angle values give us thecorresponding values of angles (A1)(L1)V and (B1)(L1)V and (C1)(L1)V and(X1)(L1)V. Knowing angle (A1)(L1)V and angle LVA equal to (L1)V(A1)gives us the value of angle V(A1)(L1). Knowing the value of all 3 anglesof the triangle and length V(A1) allows us to calculate the length ofthe sides of said triangle. Thus we can calculate the lengths of V(B1)and V(C1) and V(X1). Knowing the length of those lines and the value ofangles BAV and AVD and DVC and CVB allows us to calculate the lengths of(A1)(B1) and (B1)(C1) and (C1)(X1) and (A1)(X1). That gives us thedimensions of the truncating polygon (A1)(B1)(C1)(X1). Vertex V can alsobe truncated at an angle to the truncating base (FIG. 5), or at a planeparallel to a plane that is at an angle with the truncating base.

For the purpose of truncating vertices of a polyhedron at their realtruncating bases, the truncating bases that intersect all edges of thevertex to be truncated need to be designated as real truncating bases.In all embodiments, there are limits on truncation, those limits aredetermined by lengths of edges to be truncated and angles of truncationthat determined the real truncating bases and limits based on the amountof previous truncation at said vertex and at adjacent vertices.

The same method for truncation using the same exact steps describedabove can be used in another embodiment that can also be depicted inFIG. 1. FIG. 1 shows a polyhedron 101 constructed using a methoddisclosed herein. At a first step of the method, the three-dimensionalcoordinates of the vertices (A, B, C, D, V, W) are provided. The methodthen proceeds to a second step, where the length of every line segmentconnecting the vertices (A, B, C, D, V, W) is determined using simplealgebra. Those line segments form triangles (ABC, ABD, ABV, ABW, ACD,ACV, ACW, ADV, ADW, AVW, BCD, BCW . . . ) where some of the trianglesform triangular faces of the vertices, and the Boundary of thepolyhedron 101 (in this instance ABW, AWD, BWC, CWD, CVD, DVA, AVB, BVD,BDC), thus defining a polyhedron with a concavity in it, while the restof the triangles form polyhedron virtual triangles (including triangleBVC, which is a polyhedron virtual triangle in this instance). TriangleBVC together with triangles AWB, AWD, BWC, CWD, CVD, DVA, AVB will formwhat's defined as the Boundary Shape of this polyhedron. Then thetruncating base triangle ABC is established at adjacent vertices A, Band C that belong to the boundary shape of the polyhedron. Then thecomputations for truncation will proceed along the same computationalsequence as has been described in the previous embodiment.

Also, the same method for truncation using the same steps describedabove can be used in another embodiment that can also be depicted inFIG. 1. FIG. 1 shows a polyhedron 101 constructed using a methoddisclosed herein. At a first step of the method, the three-dimensionalcoordinates of the vertices (A, B, C, D, V, W) are provided. The methodthen proceeds to a second step, where the length of every line segmentconnecting the vertices (A, B, C, D, V, W) is determined using simplealgebra. Those line segments form triangles (ABC, ABD, ABV, ABW, ACD,ACV, ACW, ADV, ADW, AVW, BCD, BCW . . . ) where some of the trianglesform triangular faces of the vertices, and the Boundary of thepolyhedron 101 (in this instance AWB, AWD, BWC, DWC, CVD, BVD, BDC,BAD), thus defining a polyhedron with a hole in it, while the rest ofthe triangles form polyhedron virtual triangles (including triangles BVCand AVD, which are polyhedron virtual triangles in this instance).Triangles BVC and AVD together with triangles AWB, AWD, BWC, CWD, BVA,DVC will form what's defined as the Boundary Shape of this polyhedron.Then the truncating base triangle ABC is established at adjacentvertices A, B and C that belong to the boundary shape of the polyhedron.Then the computations for truncation will proceed along the samecomputational sequence as has been described in the previous embodiment.

Those skilled in the art will appreciate from the descriptions containedherein that the inventive methods and apparatus do not involve anydetermination or use of midpoints and do not involve the computations ofintersecting planes and half spaces. Further, those skilled in the artwill appreciate that properties of the polyhedra may be determined usingproperties determined for the triangles, such as surface area.

EXAMPLE

Truncating at an Angle (Q) to the Truncating Base

The following example uses FIG. 5 and truncates either at a chosen angle(Q) to the truncating base starting at point B, or at a chosen point Hon line VX with chosen distance VH. Suppose we chose the value of angleQ. The angle values we can truncate at should be less than the value ofangle VBX. On the polyhedron, we draw Line BH at an angle (Q) with lineBX, where (Q) is less than angle VBX and where H is a point on VX.Knowing angles (Q) and BXV, and length of BX in triangle BXH, wecalculate lengths of HX and HB, and determine point H. If we had startedwith a chosen point H, we would calculate angle Q and proceed to thenext step. BH intersects VL at point (L1). We need to draw a line FGwhich passes through (L1), with F being a point on AV, and G being apoint on CV. We need to determine points F and G.

We draw line FG at parallel to AC. Knowing angle (L1)BL is equal to (Q)and angle BLV, and length of BL, we calculate length of (L1)L intriangle (L1)BL. Knowing that angle V(L1)G is equal to VLC (which isequal to 180-VLA,) that angle (L1)GV is equal to ACV, and that lengthV(L1) is equal to VL-(L1)L, in triangle V(L1)G we calculate length of(LOG and length of VG and establish G. Knowing angle V(L1)F is equal toVLA, and angle (L1)FV is equal to CAV, and length V(L1) in triangleV(L1)F, we calculate length of (L1)F and length of VF and establish F.Knowing VF and VG and angle FVG in triangle FVG, we calculate length ofFG. We connect points B, F, H, G, and B to get new truncating lines BF,FH, HG and GB. Knowing angle BVA and lengths of BV and FV in triangleBVF, we calculate length of FB. Knowing angle XVA and lengths of FV andVH in triangle FVH, we calculate length of FH. Knowing angle XVC andlengths of HV and VG in triangle HVG, we calculate length of HG. Knowingangle CVB and lengths of GV and BV in triangle GVB, we calculate lengthof GB. New truncating lines BG and GH with BH form one new truncatingtriangle BGH. New truncating lines BF and FH with BH form another newtruncating triangle BFH. Those two new truncating triangles form the newtruncating polygon BGHF that determines the new face of the truncatedpolyhedron and is the new Vertex figure of Vertex V.

We can use the same method described above that established a truncatingbase triangle ABC to determine the rest of the truncating bases ofVertex V. We establish truncating base triangles BCD and ADC and DBAthat determine their own truncating bases. We can truncate at an angleto the truncating bases or parallel to the truncating bases or parallelto truncation at an angle to the truncating bases. To truncate parallelto truncation at an angle, we draw a line parallel to FG at a chosendistance and then we can proceed to do the calculations as described fortruncating in parallel. Alternatively, we can truncate vertex V at thesetruncating bases if they are the real truncating bases after wedesignate them as such. In case a truncating base is not a realtruncating base, we establish the real truncating base at an angle tothe truncating base.

EXAMPLE

Truncating a Polyhedron of Another Embodiment

The same method described above may be used to establish the value forall the intersecting angles needed to establish the truncating baseswhen the boundary shape of the Vertex to be truncated has more than fourtriangles, like for example in FIG. 6. In one embodiment that can bedepicted in FIG. 6, Vertex V has five triangular faces VAB, BVC, CVD,DVE and EVA. To establish one of its truncating bases at truncating basetriangle ABC, we need to establish the values of two intersecting angles(Q1) and (Q2). (Q1) is established since the ratio of angles ABE/EBC isequal to the ratio of angles AB(L1)/(L1)BC. Angle (L1)BC is equal toangle ABC minus angle AB(L1). Therefore, ABE/EBC is equal toAB(L1)/[ABC-AB(L1)], and angle AB(L1) may be calculated. (Q2) isestablished since the ratios of angles ABD/DBC is equal to the ratio ofangles AB(L2)/(L2)BC. Angle (L2)BC is equal to angle ABC minus angleAB(L2). Therefore, ABD/DBC is equal to AB(L2)/[ABC-AB(L2)], and angleAB(L2) may be calculated. Then we continue the steps as described abovein relation to FIG. 1 to calculate angles (Q1) and (Q2), and thenproceed to calculate the base.

Then we proceed to calculate lengths of A(L1), (L1)(L2), and (L2)C. Wehave two truncating base lines AB and BC. We calculate the lengths ofthe other three truncating base lines CZ, ZX, and XA. We truncate atbase or parallel to base or at an angle to the base etc. If we aretruncating at an angle (Y) to base, we draw BH at an angle (Y) to BX. BHintersects V(L1) at point (L3). The following takes FG parallel to AC,with F being a point on AV, and G on CV. FG passes through point L3. Weestablish points F and G following the same exact steps as for theoctahedron above. We determine length of FG. We determine point (L4)where V(L2) intersects FG using (L3)(L4)/FG is equal to (L1)(L2)/AC. Weestablish the fifth vertex I of the new truncating pentagon where B(L4)intersects VZ. We create a new truncating pentagon BGIHF formed of threenew truncating virtual triangles BGI, IBH, and HBF, whose sides we cancalculate, thus calculating the dimensions of the pentagon.

The same method described above can be used in another embodiment thatcan also be depicted in FIG. 6. FIG. 6 shows a polyhedron constructedusing method disclosed herein. At a first step of the method, thethree-dimensional coordinates of the vertices (A, B, C, D, E, V, W) areprovided. The method then proceeds to a second step, where the length ofevery line segment connecting the vertices (A, B, C, D, E, V, W) isdetermined using simple algebra. Those line segments form triangles(ABC, ABD, ABV, ABW, ACD, ACV, ACW, ADV, ADW, AVW, BCD, BCW, ABE, EVA,EVD . . . ) where some of the triangles form triangular faces of thevertices, and the Boundary of the polyhedron (in this instance AWB, AWE,EWD, BWC, CWD, BVC, BVA, CVD, AVE, BVD, EVB, EBD), defining a polyhedronwith a concavity in it, while the rest of the triangles form polyhedronvirtual triangles (including triangle DVE, which is a polyhedron virtualtriangle in this instance). Triangle DVE together with triangles AWB,AWE, EWD, BWC, CWD, BVC, BVA, CVD, AVE will form what's defined as theBoundary Shape of this polyhedron. Then the truncating base triangle ABCis established at adjacent vertices A, B and C that belong to theboundary shape of the polyhedron. Then the computations for truncationwill proceed along the same exact computational sequence as has beendescribed in the previous embodiment. Truncating this new polyhedron atan angle Y for example will give us two new truncating polygons, BGI andBHF.

This above polyhedron can have a solid part, for example, defined by thepyramid ABCDEW, and two hollow spaces defined by the pyramids BVDC andEBAV and a concavity VBED, where four faces AVB, BVC, BVD and BVE willhave edge BV in common.

We can use the same method described above that established a truncatingbase triangle ABC to determine the rest of the truncating bases ofVertex V. We establish truncating base triangles BDC and CDE and DEA andEAB that determine their own truncating bases. We can truncate at anangle to the truncating bases or parallel to the truncating bases orparallel to truncation at an angle to the truncating bases.Alternatively, we can truncate vertex V at these truncating bases ifthey are the real truncating bases after we designate them as such. Incase a truncating base is not a real truncating base, we establish thereal truncating base at an angle to the truncating base.

Optionally, one can get vertex and line length data of an object fromreal life, then display that information, and then proceed to truncatethis object using the method described above.

Using these examples above, those skilled in the art can see how totruncate an asymmetric general polyhedron of n dimensions that can beconcave and that can have a hole, using simple math.

Plotting Information on a Polyhedral Net (FIGS. 2, 3, 4)

Polyhedral nets are drawn using information gathered as set forth above.First, we draw the polyhedral net 102 of the polyhedron 101 with itstruncating base delineated (FIG. 2). We can draw the triangular faces ofthe vertex V by drawing four triangles A(V′)B, B(V)C, C(V″)D, andD(V′″)E. When the polyhedral net 102 is cut and taped together, apices(V′)(V)(V″)(V′″) will meet to form vertex V of the polyhedron. Points B,C, D represent vertices B, C, D of the polyhedron. Point E will meet atpoint A to form vertex A of the polyhedron. We modify the polyhedral netto establish a truncating base for vertex V. We define the truncatingbase ABCX by drawing truncating base lines CX and xE on faces C(V″)D andD(V′″)E respectively. X and x of the polyhedral net meet at point X ofthe polyhedron.

We can truncate at the truncating base (FIG. 3).

As shown in FIGS. 1 and 4, truncating at a new truncating plane(A1)(B1)(C1)(X1) that is parallel to truncating base ABCX gives us newtruncating lines (A1)(B1), (b1)(C1), (c1)(X1), and (x1)(a1) on thepolyhedral net. Points (B1) and (b1), (C1) and (c1), (X1) and (x1), (a1)and (A1) of the polyhedral net, meet to form points (B1), (C1), (X1),and (A1) of the polyhedron respectively.

On the net in FIG. 4, we draw a new quadrilateral face (b1)(V1)(V2)(C1)formed of two triangles (b1)(V1)(C1) and (C1)(V1)(V2) with dimensionsequal to those of the new truncating quadrilateral (A1)(B1)(C1)(X1) from(FIG. 1).

We then separate the new polyhedral net around the perimeter. Weseparate along (A)-(A1), (A1)-(B1), (B1)-(B), (B)-(b1), (b1)-(V1),(V1)-(V2), (V2)-(C1), (C1)-(C), (C)-(c1), (c1)-(X1), (X1)-(D), (D)-(x1),(x1)-(a1) and (a1)-(E) . . . and continue separating around the rest ofthe perimeter of the polyhedral net of the octahedron. We fold thefolding lines, and tape the adjacent edges together.

Implementation of and Apparatus for Practicing the Disclosed Method

FIG. 7 shows an apparatus 700 for creating and truncating generalpolyhedra without using complex formulas and without requiring advancedcomputer processors and computer memory. Apparatus 700 includes acomputer drafting station 710. Computer drafting station 710 includes aninput device 712 (e.g., a keyboard, mouse, trackball, joystick, or anyother device that may be used to input electronic data) and a display714 (e.g., a computer display, a projection device, a LCD display, acathode ray tube display, a plasma display, a LED display, or any otherappropriate display). Drafting station 710 may further include at leastone processor 715, computer memory 716, and a storage unit 718. Thestorage unit 718 may be, for example, a disk drive that stores programsand data of the computer drafting station 710. It should be appreciatedthat the computer drafting station 710 may be constructed without thestorage unit 718, though the storage unit 718 may provide additionalprogramming flexibility. The processor 715 is in data communication withthe input device 712, the display 714, and the computer memory 716. Apower supply 719 (i.e., AC power or DC power, including a battery or asolar cell, for example) is electrically coupled to the processor 715 topower the processor 715.

The storage unit 718 is illustratively shown storing (and providing tocomputer memory 716) machine readable instructions 722 a for storingvertex data input to the processor 715 via the input device 712, machinereadable instructions 722 b for implementing the method set forth aboveto truncate a polyhedron created using the vertex data, and machinereadable instructions 722 c for actuating the display 714 to presentdata. As noted above, the instructions may be contained in the computermemory 716 without use of the storage unit 718.

It is understood that while certain forms of this invention have beenillustrated and described, it is not limited thereto except insofar assuch limitations are included in the following claims and allowablefunctional equivalents thereof.

What is claimed is:
 1. An apparatus for modeling general polyhedrawithout using complex formulas and without requiring advanced computerprocessors and computer memory, the apparatus comprising: at least oneprocessor; means for inputting vertex data to the processor; a displayin data communication with the processor; and computer memory coupled tothe processor and having recorded within it machine readableinstructions for: storing vertex data previously input to the processor;determining all triangles that meet at each vertex represented by thevertex data; determining all triangular faces corresponding to eachvertex; determining polyhedron virtual triangles corresponding eachvertex; determining all polygonal faces corresponding to each vertex;modeling a polyhedron using the vertex data; and actuating the display;wherein the instructions for modeling a polyhedron utilize length dataand angle data for the determined triangles; wherein the instructionsfor modeling a general polyhedron are instructions for modeling allGeneral Polyhedra and include instructions for: establishing lines withknown lengths between the vertices; establishing triangles with knownangles from the established lines to form triangular faces of verticesof a polyhedron with some of the triangles and to form polyhedronvirtual triangles with others of the triangles; and forming polygonsthat define the faces of the polyhedron using triangular faces; whereinthe instructions for modeling a general polyhedron apply equally to: (a)a polyhedron having four faces; and (b) a polyhedron having more thanfour faces, where two or more faces share a common edge; wherein theinstructions for modeling a polyhedron apply equally to concave andconvex polyhedrons; wherein the instructions for modeling a polyhedronapply equally to polyhedra with symmetry and polyhedra with no symmetry;wherein the instructions for modeling a polyhedron apply equally topolyhedra with holes and polyhedra with no holes; wherein theinstructions for modeling a polyhedron apply equally to polyhedra thatis solid and polyhedra that is at least partially hollow; wherein themeans for inputting vertex data includes a keyboard; wherein the displayis a computer display; and wherein the apparatus further comprises meansfor powering the processor.
 2. An apparatus for creating and truncatingpolyhedra having more than four faces, the apparatus comprising: atleast one processor; means for inputting vertex data to the processor; adisplay in data communication with the processor; and computer memorycoupled to the processor and having recorded within it machine readableinstructions for: storing vertex data previously input to the processor;creating a polyhedron created using the vertex data; truncating thepolyhedron created using the vertex data; and actuating the display;wherein the instructions for creating and the instructions fortruncating utilize length data and angle data for triangles formed fromthe vertex data; wherein the instructions for creating and theinstructions for truncating include instructions for: forming twotruncating base lines for a respective vertex to be truncated;calculating the length of additional truncating base lines that completea truncating base for the vertex to be truncated; forming truncatingvirtual triangles that have sides with known lengths and angles from thetruncating base lines; and forming a truncating virtual polygon havingat least four sides from the truncating base with the truncating baselines as its sides; wherein the instructions for creating and theinstructions for truncating do not determine and do not utilize anyindication that a point is a midpoint; and wherein the instructions forcreating and the instructions for truncating do not determine and do notutilize any: (a) intersecting half plane; (b) moving plane; (c) halfspace; and (d) intersecting plane.
 3. The apparatus of claim 2, whereinthe instructions for creating and the instructions for truncatinginclude instructions for forming new truncating lines to truncateparallel to the truncating base.
 4. The apparatus of claim 2, whereinthe instructions for creating and the instructions for truncatinginclude instructions for forming new truncating lines to truncate at anangle to the truncating base.
 5. The apparatus of claim 2, wherein theinstructions for creating and the instructions for truncating includeinstructions for forming new truncating lines to truncate parallel totruncation at an angle to the truncating base.
 6. The apparatus of claim2, wherein the instructions for truncating include limitations ontruncation determined by length of edges to be truncated; angles oftruncation that determined the real truncating bases; amount of previoustruncation, and amount of truncation of adjacent vertices.
 7. Theapparatus of claim 2, wherein the instructions for truncating includeinstructions for: forming new truncating lines; establishing thedimensions of new truncating triangles with known side lengths from thenew truncating lines; and forming new truncating polygons from the newtruncating triangles.
 8. The apparatus of claim 7, wherein theinstructions for creating and the instructions for truncating includeinstructions for plotting on a polyhedral net.
 9. The apparatus of claim7, wherein the instructions for creating and the instructions fortruncating include instructions for printing the polygonal faces of thepolyhedron separate from each other.